Thank you for your prompt answer.

I do not fully understand "parental rights". I came to believe that even if a parent is not registered on the birth certificate, he still has
parental rights. In an eventual adoption procedure (or even the name change) the biological father should be notified of the procedure.

Could you please help me in understanding how can parental right be established?

Actually it is possible to change name in Italy, but you need "good reasons" (the judge decides).
In case we succeed to change his name in Italy, would this be acknowledged by the US?
thank you very much
best regards